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Properties

Label 1-6017-6017.5845-r0-0-0
Degree $1$
Conductor $6017$
Sign $-0.258 + 0.966i$
Analytic cond. $27.9428$
Root an. cond. $27.9428$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.981 − 0.192i)2-s + (−0.809 − 0.587i)3-s + (0.926 − 0.377i)4-s + (−0.998 + 0.0483i)5-s + (−0.906 − 0.421i)6-s + (−0.681 − 0.732i)7-s + (0.836 − 0.548i)8-s + (0.309 + 0.951i)9-s + (−0.970 + 0.239i)10-s + (−0.970 − 0.239i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.836 + 0.548i)15-s + (0.715 − 0.698i)16-s + (0.958 − 0.285i)17-s + (0.485 + 0.873i)18-s + ⋯
L(s)  = 1  + (0.981 − 0.192i)2-s + (−0.809 − 0.587i)3-s + (0.926 − 0.377i)4-s + (−0.998 + 0.0483i)5-s + (−0.906 − 0.421i)6-s + (−0.681 − 0.732i)7-s + (0.836 − 0.548i)8-s + (0.309 + 0.951i)9-s + (−0.970 + 0.239i)10-s + (−0.970 − 0.239i)12-s + (0.309 + 0.951i)13-s + (−0.809 − 0.587i)14-s + (0.836 + 0.548i)15-s + (0.715 − 0.698i)16-s + (0.958 − 0.285i)17-s + (0.485 + 0.873i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6017\)    =    \(11 \cdot 547\)
Sign: $-0.258 + 0.966i$
Analytic conductor: \(27.9428\)
Root analytic conductor: \(27.9428\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6017} (5845, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6017,\ (0:\ ),\ -0.258 + 0.966i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1411711291 + 0.1839055901i\)
\(L(\frac12)\) \(\approx\) \(0.1411711291 + 0.1839055901i\)
\(L(1)\) \(\approx\) \(0.9851477161 - 0.3308001257i\)
\(L(1)\) \(\approx\) \(0.9851477161 - 0.3308001257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
547 \( 1 \)
good2 \( 1 + (0.981 - 0.192i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.998 + 0.0483i)T \)
7 \( 1 + (-0.681 - 0.732i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
17 \( 1 + (0.958 - 0.285i)T \)
19 \( 1 + (-0.989 - 0.144i)T \)
23 \( 1 + (-0.748 + 0.663i)T \)
29 \( 1 + (0.995 + 0.0965i)T \)
31 \( 1 + (-0.168 + 0.985i)T \)
37 \( 1 + (0.995 + 0.0965i)T \)
41 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.970 - 0.239i)T \)
47 \( 1 + (0.215 + 0.976i)T \)
53 \( 1 + (0.779 - 0.626i)T \)
59 \( 1 + (-0.989 + 0.144i)T \)
61 \( 1 + (-0.906 - 0.421i)T \)
67 \( 1 + (-0.970 - 0.239i)T \)
71 \( 1 + (-0.168 - 0.985i)T \)
73 \( 1 + (-0.262 - 0.964i)T \)
79 \( 1 + (-0.0724 - 0.997i)T \)
83 \( 1 + (-0.998 + 0.0483i)T \)
89 \( 1 + (0.120 + 0.992i)T \)
97 \( 1 + (-0.607 + 0.794i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.100898611144216759916906316128, −16.63696846958344210016043386424, −16.11852507497932724884782779855, −15.46615383040716038634879334368, −15.051122597050279370477547968576, −14.61770948208410386000946081681, −13.38301516727092869357978907574, −12.72719245687626566032261395710, −12.23175773937643280868981972261, −11.82064453450891549102812174340, −11.07743132687899235367281330640, −10.356678679200327523019718338263, −9.87521756078154733868956487526, −8.5577777616082483386458078422, −8.17983516683751096004820038908, −7.2172164315248191904695160180, −6.40960687374078039417695249995, −5.91867401895502999167552795020, −5.349607722293017557730265318307, −4.41568026192127858773102186547, −4.001961489986038157288877924594, −3.19054974010289246863451253433, −2.65121480265848743988522231937, −1.269791121635661670720051374, −0.049889390908332564988658200991, 1.071874404499823437877001197619, 1.74791955892267477333682066716, 2.90034250888694342097868581850, 3.557524132116575228330503984024, 4.34656286534895241139333288930, 4.764009182807694528309583236890, 5.77566839444211178580015409212, 6.4926968928275650538315331647, 6.89324484215564507206449980831, 7.56571354658412438741706665464, 8.20898965503491291447495213806, 9.433433537082340248213592916291, 10.45899360100433248677248395887, 10.69094023422084835487457085265, 11.5584149813684354966649740838, 12.13489658631078601654321240041, 12.39376769103982933673439699333, 13.36710468872252738018474769174, 13.743720212226766573567144310896, 14.48367219252528390696430672438, 15.33248806375665050452859779639, 16.07777901466285986809482112372, 16.453781105957736950647837708011, 16.90344728384464776326525126309, 17.955614605700175320366429992766

Graph of the $Z$-function along the critical line